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If
$\mathrm{I}=\int \frac{\sin x+\sin ^3 x}{\cos 2 x} \mathrm{~d} x=\mathrm{P} \cos x+\mathrm{Q} \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right|+\mathrm{c},$
(where $\mathrm{c}$ is a constant of integration), then the values of $\mathrm{P}$ and $\mathrm{Q}$ are respectively
Options:
$\mathrm{I}=\int \frac{\sin x+\sin ^3 x}{\cos 2 x} \mathrm{~d} x=\mathrm{P} \cos x+\mathrm{Q} \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right|+\mathrm{c},$
(where $\mathrm{c}$ is a constant of integration), then the values of $\mathrm{P}$ and $\mathrm{Q}$ are respectively
Solution:
1520 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{2}, \frac{-3}{4 \sqrt{2}}$
$\begin{aligned} I & =\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x \\ & =\int \frac{\sin x\left(1+\sin ^2 x\right)}{2 \cos ^2 x-1} d x \\ & =\int \frac{\sin x\left(2-\cos ^2 x\right)}{2 \cos ^2 x-1} d x\end{aligned}$
$\begin{aligned} & \text { Put } \cos x=\mathrm{t} \\ & -\sin x \mathrm{~d} x=\mathrm{dt} \\ & \mathrm{I}=\int \frac{\mathrm{t}^2-2}{2 \mathrm{t}^2-1} \mathrm{dt} \\ & =\frac{1}{2} \int \frac{2 \mathrm{t}^2-4}{2 \mathrm{t}^2-1} \mathrm{dt} \\ & =\frac{1}{2}\left[\int \frac{2 \mathrm{t}^2-1}{2 \mathrm{t}^2-1} \mathrm{dt}-\int \frac{3}{2 \mathrm{t}^2-1} \mathrm{dt}\right] \\ & I=\frac{1}{2} t-\frac{1}{2} \times \frac{3}{2 \sqrt{2}} \log \left|\frac{\sqrt{2} t-1}{\sqrt{2} t+1}\right|+c \\ & =\frac{1}{2} \cos x-\frac{3}{4 \sqrt{2}} \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right|+\mathrm{c} \\ & \therefore \quad \mathrm{P}=\frac{1}{2} \text { and } \mathrm{Q}=\frac{-3}{4 \sqrt{2}} \\ & \end{aligned}$
$\begin{aligned} & \text { Put } \cos x=\mathrm{t} \\ & -\sin x \mathrm{~d} x=\mathrm{dt} \\ & \mathrm{I}=\int \frac{\mathrm{t}^2-2}{2 \mathrm{t}^2-1} \mathrm{dt} \\ & =\frac{1}{2} \int \frac{2 \mathrm{t}^2-4}{2 \mathrm{t}^2-1} \mathrm{dt} \\ & =\frac{1}{2}\left[\int \frac{2 \mathrm{t}^2-1}{2 \mathrm{t}^2-1} \mathrm{dt}-\int \frac{3}{2 \mathrm{t}^2-1} \mathrm{dt}\right] \\ & I=\frac{1}{2} t-\frac{1}{2} \times \frac{3}{2 \sqrt{2}} \log \left|\frac{\sqrt{2} t-1}{\sqrt{2} t+1}\right|+c \\ & =\frac{1}{2} \cos x-\frac{3}{4 \sqrt{2}} \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right|+\mathrm{c} \\ & \therefore \quad \mathrm{P}=\frac{1}{2} \text { and } \mathrm{Q}=\frac{-3}{4 \sqrt{2}} \\ & \end{aligned}$
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